Given $ \overrightarrow{OL}\perp\overrightarrow{ON}$, $ m \angle MON = 6x - 15$, and $ m \angle LOM = 4x - 45$, find $m\angle MON$. $O$ $L$ $N$ $M$
Answer: From the diagram, we see that together ${\angle LOM}$ and ${\angle MON}$ form ${\angle LON}$ , so $ {m\angle LOM} + {m\angle MON} = {m\angle LON}$ Since we are given that $\overrightarrow{OL}\perp\overrightarrow{ON}$ , we know ${m\angle LON = 90}$ Substitute in the expressions that were given for each measure: $ {4x - 45} + {6x - 15} = {90}$ Combine like terms: $ 10x - 60 = 90$ Add $60$ to both sides: $ 10x = 150$ Divide both sides by $10$ to find $x$ $ x = 15$ Substitute $15$ for $x$ in the expression that was given for $m\angle MON$ $ m\angle MON = 6({15}) - 15$ Simplify: $ {m\angle MON = 90 - 15}$ So ${m\angle MON = 75}$.